Multiple population-period transient spectroscopy (MUPPETS ... · MUPPETS has strong parallels to multidimensional co-herence spectroscopy (MDCS). MUPPETS measures multi-ple periods

Multiple population-period transient spectroscopy (MUPPETS) in excitonicsystemsHaorui Wu and Mark A. Berg Citation: J. Chem. Phys. 138, 034201 (2013); doi: 10.1063/1.4773982 View online: http://dx.doi.org/10.1063/1.4773982 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v138/i3 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS 138, 034201 (2013)

Multiple population-period transient spectroscopy (MUPPETS)in excitonic systems

Haorui Wu and Mark A. Berga)

Department of Chemistry and Biochemistry, University of South Carolina, Columbia,South Carolina 29208, USA

(Received 2 October 2012; accepted 11 December 2012; published online 15 January 2013)

Time-resolved experiments with more than one period of incoherent time evolution are becomingincreasingly accessible. When applied to a two-level system, these experiments separate homoge-neous and heterogeneous contributions to kinetic dispersion, i.e., to nonexponential relaxation. Here,the theory of two-dimensional (2D) multiple population-period transient spectroscopy (MUPPETS)is extended to multilevel, excitonic systems. A nonorthogonal basis set is introduced to simplifypathway calculations in multilevel systems. Because the exciton and biexciton signals have differentsigns, 2D MUPPETS cleanly separates the exciton and biexciton decays. In addition to separatinghomogeneous and heterogeneous dispersion of the exciton, correlations between the exciton andbiexciton decays are measurable. Such correlations indicate shared features in the two relaxationmechanisms. Examples are calculated as both 2D time decays and as 2D rate spectra. The effect ofsolvent heating (i.e., thermal gratings) is also calculated in multidimensional experiments on multi-level systems. © 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4773982]

I. INTRODUCTION

Kinetic measurements are a major subset of physicalchemistry and take on many different forms appropriate todifferent processes and timescales. Nonetheless, almost allare one dimensional (1D): a single period of time exists be-tween a single perturbation of the system and a later detectionof its evolved state. Our group has been exploring multidi-mensional kinetics in which there is more than one pertur-bation, and thus, more than one period of time evolution.1–10

We have called our approach, which uses weak optical per-turbations, multiple population-period transient spectroscopy(MUPPETS). So far, its focus has been on nonexponentialrelaxation (rate dispersion) in two-level systems. In thosesystems, MUPPETS can separate homogeneous and heteroge-neous contributions to rate dispersion. This paper lays a the-oretical foundation for MUPPETS in multilevel systems andespecially in excitonic systems—those with equally spacedlevels and optical transitions and relaxations that occur insingle steps. The most important new features are the abil-ity to accurately separate exciton and biexciton dynamics andto measure correlations in the rate dispersion of exciton andbiexciton relaxation. Related experimental results on excitonand biexciton dynamics in CdSe nanoparticles will be pub-lished in the near future.11, 12

The concept of MUPPETS is illustrated in Fig. 1. Atotal of six pulses are used: three (1–3) simultaneous pairs(a and b) separated by two times, τ 1 and τ 2. Each pair causesan incoherent transition, i.e., a transition from one quantummechanical population to another. Any coherence is assumedto be quenched by rapid dephasing. The novel aspect ofMUPPETS is that the correlated relaxation of the population

a)Author to whom correspondence should be addressed. Electronic mail:[email protected].

during two time periods is measured. Neither ensemble av-eraging nor relaxation of the molecule occurs between theseperiods, so different processes are accessible than in experi-ments with only one relaxation time. Understanding the re-sulting multidimensional correlation functions when severalpopulation states are accessible is a primary aim of this paper.

The pulses in each pair come from different directions,so the populations consist of spatial gratings.13–15 Detectionis by diffraction of pulse 3a from the final population gratingand heterodyning the diffracted light with pulse 3b. (Practicaldetection schemes also account for diffraction in the oppo-site direction.4) As Fig. 1(a) suggests, it is possible to arrangethe phase-matching geometry such that diffraction only oc-curs from planes created by the combined action of all fourexcitation pulses. These more technical aspects of the experi-ment will not be treated here. It is only important to know thatit is practical to isolate a signal that is confined to exactly oneelectric-field interaction with each of the six pulses.

As with 1D kinetics, theoretical concepts transcend thevarious experimental implementations needed for differenttimescales and processes. In existing experiments, MUPPETShas focused on electronic-state relaxation on subnanosecondtimescales. However, the theoretical ideas developed here areequally applicable to any timescale. With modest modifica-tion, they can also find application to other types of perturba-tion and other relaxation processes.

MUPPETS has strong parallels to multidimensional co-herence spectroscopy (MDCS). MUPPETS measures multi-ple periods of incoherent evolution (kinetic rates), whereasMDCS measures multiple periods of coherent evolution(spectral frequencies). MDCS began with two-level systems,in which they give “echo” phenomena.16, 17 These exper-iments separate homogeneous and inhomogeneous contri-butions to spectral linewidths, just as MUPPETS of two-

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034201-2 H. Wu and M. A. Berg J. Chem. Phys. 138, 034201 (2013)

FIG. 1. Schematic of the MUPPETS experiment. (a) The upper and lower panels represent rapidly and slowly relaxing subensembles within the sample. Twosimultaneous pulses (1a and 1b) from different directions intersect in the sample to create a spatial grating of excited-state molecules (red). After a time τ 1, asecond pair of pulses (2a and 2b) create a second grating of excited molecules (blue). The slow subensemble now contains vertical diffraction planes formedby regions that have interacted twice (black), once (red and blue) and never (white). After an additional time τ 2, pulse 3a is diffracted from these planes and iscombined with pulse 3b for heterodyne detection. The diffraction isolates the signal unique to one interaction with the first excitation and one interaction withthe second excitation. (b) An accurate representation of the pulse directions used in the experiment: tan – lens, orange – sample.

level systems separates homogeneous and heterogeneousrate dispersion. When MDCS was extended to multilevelsystems, it became various forms of spectral correlationspectroscopy, which reveal coupling between different spec-tral transitions.18–22 MDCS is well established in nuclearmagnetic resonance18, 19 and, more recently, has been ex-tended to electronic20 and vibrational21, 22 transitions. In thelatter two forms, it has been especially valuable in excitonicsystems,23–29 where the transitions are strongly overlapped in1D spectra. By analogy, one anticipates that MUPPETS inmultilevel systems will probe correlations in the relaxation ofdifferent transitions and will be especially relevant in exci-tonic systems, where spectral discrimination of different tran-sitions can be difficult.

One goal of the paper is to clarify the meaning of theintertransition correlations that we anticipate. Another is toillustrate the interplay of the intertransition and intratransitioncontributions to the total experimental signal. To tackle theseproblems, we first develop simplified methods for calculatingmultidimensional incoherent signals in excitonic systems andthen use them to calculate results for several simple, limitingmodels.

In two-level systems, it is common to reduce the dimen-sionality of the problem by changing the basis set. The totalpopulation is invariant, and only the dynamics of the popula-tion difference need to be calculated. The primary simplifica-tions in the current calculations come from extending this ideato multilevel systems. A nonorthogonal coordinate system isrequired, but this feature is easily handled by the Hilbert-space formalism that we introduced previously.5, 6 The pri-mary new difficulty in multilevel systems is the possibilityof cross-relaxation between basis states. Fortunately, this ef-fect is minimized when higher excitons relax faster than lowerexcitons. This situation is common due to processes that arecalled exciton–exciton annihilation in molecular systems orAuger relaxation in semiconductors. Approximations for thiscase are found. Section II develops the general formalism, andthen Sec. III looks in more detail at two-dimensional (2D)MUPPETS for several different energy-level schemes.

These results lead to several useful results that are ex-plored in Sec. IV. Separating exciton and biexciton kinet-

ics can be difficult when the spectral exciton shift is small.MUPPETS is a sensitive and robust method for separatingexciton and biexciton dynamics that does not rely on spec-tral separation. It is also insensitive to the formation of pho-toproducts, which can complicate power-dependent measure-ments. In general, the level of coupling between zero-orderchromophores needed to create an exciton for purposes ofMUPPETS (an incoherent exciton) is much lower than thatneeded to create an exciton for purposes of coherent spec-troscopy (a coherent exciton). Thus, MUPPETS can be usefulfor studying weakly coupled systems.

Example calculations are presented on four model sys-tems with identical 1D kinetics in Sec. IV C. These modelsmix homogeneous and heterogeneous exciton relaxation withbiexcitons that are either correlated or uncorrelated with theexciton relaxation. Despite having identical 1D kinetics anddespite the overlap of intra- and intertransition features, eachmodel produces very different 2D results and would be readilydistinguishable in a 2D MUPPETS experiment. Rate correla-tion between different transitions is shown to be analogous tohomogeneous kinetics on a single transition. Correlation be-tween exciton and biexciton relaxation is possible whether ornot the individual transitions are homogeneous or heteroge-neous. Intertransition rate correlations indicate a shared fea-ture in the two relaxation mechanisms such as dependence ona common bath mode.

Real MUPPETS experiments detect not only the res-onant signal due to the chromophore, but also see solventheating due to chromophore relaxation.7 These thermal ef-fects are the multidimensional extension of thermal-gratingspectroscopy.13–15 They are both a complication to measur-ing the resonant signal and a potential route to measuringnonradiative relaxation between spectroscopically dark states.The theory needed to calculate thermal effects in multilevelMUPPETS experiments is developed in Sec. V.

II. THEORY FOR MULTISTATE SYSTEMS

The Hilbert-space pathway formalism for calculatingmultidimensional incoherent experiments has been discussedin detail previously.5, 6, 10 In this formalism, as the number



of states in the system increases, the number of pathwaysincreases combinatorially. This section seeks to simplify suchcalculations. Section II A summarizes previous Hilbert-spaceresults in a convenient notation. Section II B introduces a newbasis set to simplify these calculations in a general multistatesystem. Section II C then specializes to excitonic systems,which will be the focus of the remainder of the paper.

A. Review of incoherent Hilbert-space calculations

The signal from an N-dimensional heterodynedexperiment is the change in fluence of the (N+1)th (lo-cal oscillator) beam δIN+1(�) relative to its total fluence IN+1,as a function of the local-oscillator–probe phase difference�. This change can be expressed as an absorbance A(N)(�;τN, . . . , τ 1),

A(N)(�; τN, . . . , τ1) = (−1)NδIN+1(�)

IN+1, (1)

where τ n is the time interval between pulses n and n+1.Fourier transforming the phase-dependence extracts a com-plex absorbance A(N)(τN, . . . , τ 1), which obeys a generalizedBeer’s law,7, 10

A(N)(τN, . . . , τ1) = (−1)N ρL 〈σD〉◦ . (2)

This expression contains the detection cross-section operatorσD , the number density of solute molecules ρ, and the lengthof the sample L.

The expectation value of σD is calculated as a matrix el-ement in the incoherent Hilbert space,

〈σD〉◦ = [I |σD|f (τN, . . . , τ1)]◦, (3)

where [I| is the identity state [see Eq. (17)] and | f ] is the finalstate of the system at the time of detection. The degree sign in-dicates that the calculations are done without the phase factorsfor the excitation fields.7 The phase convention for the com-plex absorbance is the same as for the complex cross-section:real parts correspond to absorption; imaginary parts corre-spond to index-of-refraction. The final state | f ] is obtainedfrom the initial, equilibrium state |eq] by successive operatorsTn, representing optical transitions due to the nth excitation,and G(tn, tn−1), representing evolution between times tn−1 andtn,

|f (τN, . . . , τ1)]◦ = G(tN , tN−1)T◦N . . . G(t1, t0)T◦

1|eq]. (4)

Throughout the paper, absolute times will be denoted tn, andtime intervals will be given by

τn = tn − tn−1. (5)

The equation of motion for an arbitrary state |P] containsthe rate operator R(t)

d

dt|P ] = −R(t)|P ]. (6)

For nonexponential relaxations, the rates are time dependent.The Green’s operator G(tn, tn−1) is then nonstationary

G(t2, t1) = exp+

(−

∫ t2

t1

R(t)dt

), (7)

where the exponential is time ordered.30

The optical-transition operator T◦n is given by

T◦n =

∑i,j∈{a,b}

In,ijσT Kn,ij ( �M · ��n,ij ). (8)

The nth excitation consists of two pulses labeled a and b (seeFig. 1), and in Eq. (8), the sum runs over the four permuta-tions of these pulses. The effective fluence of the pair In,ij isthe geometric mean of the fluences of the two pulses, In,i andIn,j: In,ij = (In,iIn, j)1/2. The transition cross-section operator σT

is constructed from the absorption cross-sections of the elec-tronic transitions of the system. Unlike the detection cross-section σD , which is complex, σT has only real elements.The dipole-moment tensor �M and the polarization tensor ��n,ij

are required to calculate the effects of chromophore rota-tion, but will be neglected in this paper. The phase-matchingconditions are generated by the grating-vector operator Kn,ij .We assume that one combination of pulses is perfectly phasematched, and all others are poorly phase matched.

With these assumptions, the equation for the signal re-duces to

A(N)(τN, . . . , τ1) = (−1)N ρLI (N) [I |σDG(tN , tN−1) . . .

×σT G(t1, t0)σT |eq], (9)

with

I (N) = IN,ab · · · I1,ab (10)

representing the total excitation fluence from N pulse pairs. Inthe case where every pulse has the same fluence I, I(N) = IN.The next step is to introduce complete sets of states betweeneach pair of operators in Eq. (9). The results are more compactif we adopt the notation

[n|O|m] = Omn (11)

for the matrix element of an operator O between states [n|and |m]. Assuming summation over repeated indices, Eq. (9)reduces to

A(N)(τN, . . . , τ1)

ρLI (N)= (−1)N

⟨(σD)nI Gm

n (tN , tN−1) . . .

×(σT )jk Gij (t1, t0) (σT )eqi

⟩. (12)

Each term in the implied sum represents one Hilbert-spacepathway. This sum is calculated for a single chromophore be-fore averaging over the ensemble, as indicated by the angularbrackets.

If the optical cross-sections are independent of time, thetime dependence and relative weight of each pathway can beseparated

A(N)(τN, . . . , τ1)

ρLI (N)= (−1)N B

n,...,j,eq

I,...,k,i Cm,...,in,...,j (τN, . . . , τ1).

(13)Each pathway is defined by the set of intermediate states{i, . . . , n}. The dynamics associated with a pathway are givenby the correlation-function matrix

Cm,...,in,...,j (τN, . . . , τ1) = ⟨

Gmn (tN , tN−1) . . . Gi

j (t1, t0)⟩. (14)

Each element of this 2N-dimensional matrix is an N-time-interval correlation function. Each correlation function is the



ensemble average of N time-evolution operators. The relativeweight of each pathway is given by

Bn,l,...,eq

I,m,...,i = (σD)nI . . . (σT )jk (σT )eqi . (15)

Because two of its indices are fixed, this matrix also has 2Ndimensions. Each element gives the total cross-section of thecorresponding element of the correlation-function matrix. Thescalar product of these two matrices in Eq. (13) sums the cor-relation functions from all the pathways with their appropriatecross-sections.

B. Basis set to reduce the dimensionalityof the problem

Here, we consider the general problem of a good basis setfor pathway calculations in a system with N optical levels,{|0], |1], . . . , |N−1]}. It is desirable to have the initial, equi-librium state |eq] as one member of the basis set. If the statespacing is large, only the lowest state is occupied in equilib-rium: |0] = |eq]. It is also desirable to have the identity state[I| as a member of the basis set. Thus, in the new, primed basisset, {|0′], |1′], . . . , |N−1′]}, we require

|0′] = |eq] = |0] (16)

and

[0′| = [I | =N−1∑n=0

[n|. (17)

With these conditions, all pathways begin with |0′] and endwith [0′| [see Eqs. (9) and (12)].

An orthogonal basis set cannot satisfy both Eqs. (16) and(17). However, in a nonorthogonal basis, bras and kets neednot be identical: they are described by different, dual basissets.31 In such a nonorthogonal system, the nonzero kets mustbe orthogonal to [0′|

[0′|n′] = [I |n′] = δ0,n. (18)

Because the identity state measures the total population of astate,5 Eq. (18) means that the nonzero primed kets do nothave any net population: they consist only of population dif-ferences. As a result, the rate operator R(t) cannot connectthe zero and nonzero kets without changing the total popula-tion of the system. These two sets of states, zero prime andnonzero prime, are the irreducible sets resulting from the lawof population conservation. In addition, |0′] = |eq] cannot de-cay; it is unaffected by R(t). Thus, it is possible to reduce thedimensionality of the rate matrix Ri ′

j ′(t) by eliminating its 0′

row and column.The exact form of the nonzero states has not been spec-

ified. We choose the first excited state [1′| so it is the onlynonzero transition out of |0′]

[n′|σT |0′] = δn′1′ (σT )0′1′ . (19)

The transition cross-section operator acts on a general state|P] in a perturbative fashion6

(1 + λσT ) |P ] = |P ] + λσT |P ]. (20)

By the conservation of population, σT acting on any state canonly create a new state with no population, that is, a superpo-sition of nonzero primed states. Thus,

[0′|σT |n′] = 0. (21)

With Eqs. (19) and (21), the transition cross-section matrix(σT )ij can also be reduced in dimension by eliminating its 0′

row and column.This procedure drops one nonzero element (σT )0′

1′ , whichoccurs on the first step in every pathway. The effect of this el-ement will be included in a new detection vector [σ D|, whichis defined by

[σD| = (σT )0′1′

Re (σD)0′0′

[0′|σD. (22)

Because all pathways end on [0′|, the detection matrix andfinal state can be replaced by this vector. Because there are notransitions into |0′], the first element of the detection matrixonly occurs in static (N = 0) spectroscopy

A(0) = ρL (σD)0′0′ . (23)

For all higher order measurements, the n = 0 element of(σ D)n ′

can be dropped, and the detection vector can be re-duced in dimension. The term Re (σD)0′

0′ is included in the def-inition of [σ D| for convenience: Using Eq. (23), the Nth-orderabsorbance will scale explicitly with the static absorbanceA′(0).

Equation (13) can now be re-expressed as

A(N)(τN, . . . , τ1)

A′(0)= I (N)σ

n′,...,l′,j ′m′,...,k′ C

m′,...,k′,1′n′,...,l′,j ′ (τN, . . . , τ1),

(24)for N �= 0. The total cross-section,

σn′,...,l′,j ′m′,...,k′ = (−1)N (σD)n

′(σT )l

′m′ . . . (σT )j

′k′ , (25)

gives the relative weight of each pathway, but is a lowerdimensional matrix than B

n,...,j,eq

I,...,k,i [Eq. (15)]. It contains Ncross-sections to match the N fluence factors in I(N). The cor-relation function is also simplified relative to Eq. (14), be-cause its first index is now fixed. Equation (24) generalizes afamiliar expression for the fractional population change in apump–probe experiment,

A′(τ )

A′0

= ρ(τ )

ρ0= −IσC(τ ). (26)

The indices in Eq. (24) only run over nonzero values.Thus, in the primed basis set, the entire calculation is re-stricted to nonzero intermediate states, and the problem is re-duced by one dimension. The reduction is possible because ofthe restrictions implied by population conservation.

For a two-level system, Eqs. (16) and (18) completelydetermine the primed basis set,

|0′] = |0],

|1′] = 1√2

(|1] − |0]) ,(27)

and its dual basis set,

[0′| = [1| + [0|,(28)

[1′| =√

2[1|.



It is also possible to include population conservation in a two-level system by using an orthogonal basis set.6 Either ap-proach is viable, but the current one generalizes to multilevelsystems.

C. Application to excitonic systems

For more than two states, Eqs. (16) and (18) do not com-pletely define the higher basis states. Choices can be made tofurther simplify the transition and rate matrices, but more de-tailed knowledge of the structure of these matrices is needed.We specialize to excitonic systems, which are defined as a setof equally spaced states or groups of nearly degenerate statesthat undergo optical transitions and relaxation in incrementsof one “quantum” at a time. The transition and rate matricesof an excitonic system are simplified if the nonzero primedbasis kets are chosen to be differences of neighboring states,

|n′] = 1√2

(|n] − |n − 1]) , n ≥ 1, (29)

with the dual states

[n′| =√

2N∑

i=n

[i|, n ≥ 1. (30)

The remainder of the paper focuses on 1D and 2D exper-iments. These experiments cannot access states higher than|3], so four-level schemes will be sufficient. The standard ba-sis set for such schemes is {|3], |2], |1], |0]} (triexciton, biex-citon, exciton, and ground states, respectively). The same ratematrix applies for all schemes,

Rij (t) =

⎛⎜⎜⎝

kt (t) 0 0 0−kt (t) kb(t) 0 0

0 −kb(t) ke(t) 00 0 −ke(t) 0

⎞⎟⎟⎠ , (31)

where kt(t) is the triexciton-to-biexciton rate, kb(t) is thebiexciton-to-exciton rate, and ke(t) is the exciton-to-ground-state rate. When transformed to the primed basis set, the ratematrix becomes

Ri ′j ′ (t) =

⎛⎜⎜⎝

kt (t) 0 0 0−kb(t) kb(t) 0 0

0 −ke(t) ke(t) 00 0 0 0

⎞⎟⎟⎠ , (32)

which can be reduced in dimensionality to

Ri ′j ′ (t) =

⎛⎝ kt (t) 0 0

−kb(t) kb(t) 00 −ke(t) ke(t)

⎞⎠ . (33)

In addition, the total signal given by Eq. (24) simplifiesbecause the first excited state defined by Eq. (19) is also thelowest state in the relaxation scheme given by Eq. (33), thatis, j = 1′

A(N)(τN, . . . , τ1)

A′(0)= I (N)σ

n′,...,l′,1′m′,...,k′ C

m′,...,k′,1′n′,...,l′,1′ (τN, . . . , τ1).

(34)Thus, the signal is calculated as a product of two 2(N−1)-dimensional matrices, one dimension lower than in Eq. (13).

FIG. 2. Three energy-level schemes for an excitonic system. Red arrows areallowed optical transitions with each arrow indicating a factor of σ in cross-section. Blue arrows indicate nonradiative transitions; dashed arrows are fastrelaxations.

Off-diagonal elements in the rate matrix add complexityto the calculations. It is not generally possible to diagonalizethe rate matrix with any coordinate transformation. However,in the primed basis set, the off-diagonal terms become smallif each higher exciton relaxes rapidly compared to lower ex-citons. As discussed in Sec. IV B below, this limit can be re-garded as one of strong incoherent coupling. In the currentexample,

Ri ′j ′ (t)

kt �kb�ke−−−−−−−→⎛⎝ kt (t) 0 0

0 kb(t) 00 0 ke(t)

⎞⎠ . (35)

The transition and detection cross-section matrices de-pend on the spectroscopic details of the system. Three ex-amples are shown in Fig. 2. They have been chosen toillustrate important limiting behaviors in the final signal.Scheme A represents an exciton consisting of many coupledchromophores (M → ∞, see Sec. IV B). The ground-to-exciton transition has the same cross-section as the exciton-to-biexciton and biexciton-to-triexciton transitions: σ 01 = σ 12

= σ 23 = σ . In addition, the downward transitions have thesame cross-section as the upward transitions: σ 01 = σ 10, σ 12

= σ 21 and σ 23 = σ 32.Alternatively, the exciton levels may not be eigenstates.

They may have internal structure or dynamics within a bandof nearly degenerate eigenstates. Scheme B is an exam-ple. Absorption to a bright state is followed by rapid relax-ation to a state with zero emission cross-section: σ 10 = σ 21

= 0. The ground-to-exciton and exciton-to-biexciton transi-tions still have the same strength: σ 01 = σ 12 = σ . No triexci-ton state is included.

Scheme C is similar to Scheme B, in that it has no triexci-ton and no emission (σ 10 = σ 21 = 0). However, it consists offew coupled chromophores, so the exciton-to-biexciton transi-tion has a lower cross-section than the ground-to-exciton tran-sition. We choose σ 01 = 2σ 12 = 2σ (M = 2, see Sec. IV B).CdSe nanoparticles with band-edge excitation are a real sys-tem approximated by model C.11, 12, 32



For Scheme A in the standard basis set, the transition ma-trix is

(σT )ij = σ ′

⎛⎜⎜⎝

−1 1 0 01 −2 1 00 1 −2 10 0 1 −1

⎞⎟⎟⎠ , (36)

and the detection matrix is

(σD)ij = σ

2

⎛⎜⎜⎝

−1 1 0 0−1 0 1 00 −1 0 10 0 −1 1

⎞⎟⎟⎠ , (37)

where the prime indicates the real part of the complex cross-section. In the primed basis set, these matrices become

(σT )i′

j ′ = σ ′

⎛⎜⎜⎝

−2 1 0 01 −2 1 00 1 −2

√2

0 0 0 0

⎞⎟⎟⎠ (38)

and

(σD)i′

j ′ = σ

2

⎛⎜⎜⎝

−2 1 0 0−3 0 1 0−2 −1 0

√2

−√2 0 −√

2 2

⎞⎟⎟⎠ . (39)

Reducing the dimensionality of the matrices yields

(σT )i′

j ′ = σ ′

⎛⎝−2 1 0

1 −2 10 1 −2

⎞⎠ (40)

and

(σD)i′ = σ

(−1 0 −1). (41)

To evaluate the total signal for Scheme A, Eqs. (40)and (41) are inserted into Eq. (25) and evaluated by stan-dard matrix methods to yield the relative cross-section of eachpathway σ

n′,...,l′,1′m′,...,k′ . The correlation function for each pathway

Cm′,...,k′,1′n′,...,l′,1′ (τN, . . . , τ1) is evaluated by putting Eq. (33) into

Eqs. (7) and (14). These components are put into Eq. (34)to give the experimental signal. Examples of this procedureare given in Sec. III.

III. PATHWAY CALCULATIONS IN EXCITONICSYSTEMS

A. Cross-sections

In the standard basis set, Eq. (13) yields three pathwayswith nonzero amplitude for 1D experiments and 16 pathwaysfor 2D experiments. In the primed basis set using Eq. (34),the number of pathways is reduced to one for 1D experimentsand three for 2D experiments. These pathways are shown onthe right-hand side of Fig. 3. Each pathway is represented asa series of transformation from the initial state on the right tothe final state on the left. Each transformation is representedas an arrow and contributes a matrix element of the operatorgoverning the transformation, which is shown below the path-ways. The final state of each pathway is detected by form-ing the product with the detection vector [σ D|. The strong

FIG. 3. Pathways for the calculation of one-dimensional (1D) and two-dimensional (2D) signals. The right-hand side shows the allowed pathwaysbetween states |n′] in the primed basis set. The operators responsible for eachtransition are given below the arrows: G, the time-evolution operator and σT ′the optical transition operator. The indices corresponding to each level in thepathway are indicated above the solid line. The final state in each pathwayis detected by taking the product with the detection vector [σD|. The totalcross-section for each pathway is given in the center of the figure for each ofthe energy-level schemes shown in Fig. 2. The correlation function for eachpathway is given on the left. Pathways (i) and (iii) have only diagonal relax-ation and dominate when the biexciton decay is faster than the exciton decay.Pathway (ii) (gray) involves cross-relaxation and is a minor contribution.

selection rules in the primed basis set allow one to quicklyenumerate the pathways with nonzero amplitude on suchdiagrams.

The correlation function corresponding to each pathwayis shown on the left-hand side of Fig. 3. It is formed fromthe matrix elements of the time-evolution operator of the cor-responding pathway through Eq. (14). The steps in the path-ways are labeled above the solid line with the indices used inour equations.

In the case where biexciton relaxation is faster than ex-citon relaxation, pathway (ii), which is in gray, has onlya small contribution. That pathway will be discussed inSec. III C. For now, we only consider the two dominant 2Dpathways. Note that the triexciton state contributes to the de-tection cross-section in Scheme A, but |3′] cannot occur as anintermediate state in a 2D experiment.

The total cross-section for each pathway is calculatedfrom the matrix elements of the cross-section operators, σT

and σD , according to Eq. (25). The exact cross-section foreach pathway and, in particular, the relative contributions ofexciton and biexciton dynamics, depends on the details ofthe state scheme. Results for the three schemes of Fig. 2 areshown in the center of Fig. 3. Scheme A is a limiting case(see Sec. IV B below) where excitons are detectable and biex-citons are not. The only pathway involving biexciton dynam-ics, pathway (i), has a cross-section of zero. Scheme B is inthe opposite limit: biexcitons are directly detectable and ex-citons are not. Scheme B gives no signal in a 1D experiment,and 2D pathway (iii) has a cross-section of zero. However, 2Dpathway (i) has a nonzero cross-section and can be measuredin Scheme B. Information on both exciton and biexciton dy-namics are available from this pathway.

Scheme C is an intermediate case where pathways end-ing with either excitons or biexcitons contribute to the signal.The notable feature is that the two pathways (i) and (iii) have



opposite signs. Generally, the biexciton relaxes faster thanthe exciton, and the signal will initially rise as the negativebiexciton signal decays. This feature allows 2D MUPPETS tocleanly separate exciton and biexciton dynamics, as will beillustrated in Sec. IV A.

To summarize, the relative contributions of exciton andbiexciton dynamics to a 2D experiment vary with the tran-sition cross-sections of the system of interest. These cross-sections determine both the relative signs and magnitudes ofthe correlation functions that are measured, and thus, the typeof dynamical information that is available.

B. Diagonal correlation functions

The reduction in the number of pathways in the primedbasis set not only simplifies the calculation of the amplitudes;it also reduces the number of correlation functions to a mini-mum. Figure 3 shows that a 1D experiment is described by asingle correlation function C1′

1′ (τ1). This correlation functionis diagonal in the sense that in one time period it only mea-sures survival of one basis state. In this case, the notation canbe simplified: Ci

i (τ1) = Ci(τ1). This type of correlation func-tion is normalized to one at the time origin.

Using Eqs. (7), (14) and (33), the exciton decay measuredin a 1D experiment is given by

C1′(τ1) = ⟨G1′

1′ (t1, t0)⟩

=⟨exp

(−

∫ t1

t0

ke(t)dt

)⟩. (42)

A similar correlation function,

C2′ (τ1) = ⟨G2′

2′(t1, t0)⟩

=⟨exp

(−

∫ t1

t0

kb(t)dt

)⟩, (43)

defines the biexciton decay, but it cannot be measured in a 1Dexperiment.

The 2D signals are dominated by diagonal correlationfunctions. The exciton–exciton correlation function,

C1′1′ (τ2, τ1) = ⟨G1′

1′ (t2, t1)G1′1′ (t1, t0)

⟩

=⟨exp

(−

∫ t2

t1

ke(t)dt −∫ t1

t0

ke(t)dt

)⟩, (44)

occurs in pathway (iii) of Fig. 3. It is essentially similar tothe 2D correlation function previously studied in two-levelsystems.3, 8, 10 If the decay is nonexponential due to homoge-neous causes, the 2D correlation function is the product of 1Dcorrelation functions,

C1′1′(τ2, τ1) = C1′(τ2)C1′(τ1). (45)

If the decay is heterogeneous, the 2D correlation function isequal to the 1D correlation function of the sum of the timevariables,

C1′1′ (τ2, τ1) = C1′(τ2 + τ1). (46)

Thus, with 2D MUPPETS in an excitonic system, it is pos-sible to distinguish homogeneous and heterogeneous mecha-

nisms of rate dispersion of the exciton decay, just as it is in atwo-level system.

A new feature of MUPPETS in multilevel systems is thepossibility of cross-correlations between different relaxations.For example, pathway (i) in Fig. 3 has an exciton–biexcitoncorrelation function,

C2′1′ (τ2, τ1) = ⟨G2′

2′(t2, t1)G1′1′(t1, t0)

⟩

=⟨exp

(−

∫ t2

t1

kb(t)dt −∫ t1

t0

ke(t)dt

)⟩. (47)

Although two transitions are involved, the correlation is stilldiagonal during each time interval. When τ 1 = 0, this functiongives access to the biexciton decay [Eq. (43)],

C2′1′ (τ2, 0) = C2′(τ2). (48)

More generally, C2′1′(τ 2, τ 1) is sensitive to correlations be-tween exciton and biexciton dynamics. These correlations arean important new feature in multilevel MUPPETS and are il-lustrated with examples in Sec. IV C.

C. Off-diagonal correlation functions

In addition to the diagonal correlation functions just dis-cussed, multilevel systems also have correlations involving re-laxation between basis states during one of the time periods.These correlation functions involve off-diagonal elements ofthe rate matrix. For example, pathway (ii) in Fig. 3 has thecorrelation function

C2′1′1′ (τ2, τ1) = ⟨

G2′1′(t2, t1)G1′

1′(t1, t0)⟩, (49)

which involves relaxation from |2′] to |1′] during τ 2. The Ap-pendix [see Eq. (A5)] shows that the off-diagonal time evolu-tion can be calculated exactly once a dynamic model for thediagonal elements is specified

G2′1′ (t2, t1) =

∫ t2

t1

G1′1′ (t2, t

′)k1′(t ′)G2′2′ (t ′, t1)dt ′. (50)

However, it is difficult to make general statements about thefull correlation function from this exact expression.

Fortunately, the primed basis set makes the cross-relaxation small when biexciton relaxation is faster than ex-citon relaxation. In this case, the relaxation of the standardbasis state |2] is biphasic: first |2] decays to |1], and then |1]decays to |0]. In the primed basis, this decay is representedby a sum of G2′

2′ (t2, t1) and G1′1′ (t2, t1). However, this sum con-

tains a small error: the decay of |1] does not start immedi-ately as it does in G1′

1′ (t2, t1); the start of its decay is delayedby the time needed for the biexciton to decay. This correc-tion is isolated as the off-diagonal time evolution G2′

1′ (t2, t1).If the decay of the exciton during the biexciton lifetime issmall, the correction is small. The Appendix shows that in thislimit, the off-diagonal time evolution can be approximatedby [see Eq. (A11)]

G2′1′ (t2, t1) = G2′

2′ (t2, t1)(1 − G1′1′(t2, t1)). (51)

The cross-relaxation correlation function cannot be cal-culated until the correlation between exciton and biexciton



dynamics are specified. However, its properties can be illus-trated with the case of uncorrelated dynamics. In that case,the 1D and 2D correlation cross-relaxation correlation func-tions can be expressed in terms of the diagonal correlationfunctions,

C2′1′ (τ1) = ⟨

G2′1′ (t2, t1)

⟩

= C2′(τ1) (1 − C1′(τ1)) (52)

and

C2′1′1′ (τ2, τ1) = C2′(τ2) (C1′(τ1) − C1′1′(τ1, τ2)) . (53)

Cross-relaxations are not normalizable: they are zero atthe time origin. Their contribution to the signal must bejudged not by their cross-section, as given in Fig. 3, but bytheir maximum size. The 2D function C2′

1′1′ (τ2, τ1) is zerowhenever τ 2 = 0. Its maximum lies along τ 1 = 0, where itis equal to the 1D cross-relaxation function,

C2′1′1′ (τ2, 0) = C2′

1′ (τ2). (54)

It rises slowly in τ 2 with the exciton decay C1′ (τ 2), but iscut off by the rapid biexciton decay C2′(τ 2) [see Eq. (52) andFig. 4(a)]. If the dynamics can be characterized by averagerate constants, the maximum value of C2′

1′ (τ2) is approxi-mately ke/kb.

FIG. 4. The 1D kinetics used in the example calculations (Figs. 5–7), whichare identical for all the models. (a) Time decays: exciton decay C1′ (τ ) [upper,red curve, Eq. (55)], biexciton decay C2′ (τ ) [middle, blue curve, Eq. (56)],and cross-relaxation C2′

1′ (τ1) [lowest, green curve, Eq. (52)]. (b) Rate spec-tra: Exciton spectrum C1′ (y) (rightmost, red curve) and biexciton spectrumC2′ (y) (leftmost, blue curve) with y = ln(κτ 0).

IV. EXAMPLES OF NEW EFFECTS

A. Separating exciton and biexciton dynamics

This section will present calculations of 2D-MUPPETSresults for several simple models of the dynamics. All themodels are based on state Scheme C in Fig. 2, where all path-ways are active. The 1D correlation functions for all the ex-amples will be the same

C1′(τ1) = exp[− (τ1/τ0)1/2] (55)

for the exciton and

C2′ (τ1) = exp[− (10τ1/τ0)1/2] (56)

for the biexciton. These two decays are similar,

C2′(τ1) = C1′(cτ1), (57)

with the biexciton decaying ten times faster (c = 10) than theexciton.

The decays are stretched exponentials in time and areshown in Fig. 4(a). The cross-relaxation C2′

1′ (τ1) in the un-correlated limit [Eq. (52)] is also shown in Fig. 4(a). As ex-pected, the large difference between exciton and biexciton de-cay times makes this term small.

In addition to the time-domain decays, it is useful to lookat rate spectra. The rate spectrum C(y) of a correlation func-tion C(τ ) is defined implicitly by

C(τ ) =∫ ∞

−∞C(y) exp(−τey/τ0)dy. (58)

The rate spectrum is essentially the inverse Laplace trans-form of the time decay expressed on a logarithmic scale,y = ln(κτ 0), where κ is the Laplace rate. More detail onthe properties and calculation of rate spectra can be found inRef. 2. The rate spectrum C1′(y) of the stretched exponentialin Eq. (55) is shown in Fig. 4(b). Applying the transform inEq. (58) twice, a 2D time function C(τ 2, τ 1) can be expressedas a rate-correlation spectrum C(y2, y1).

The experimental signal in a 1D experiment is directlyrelated to the 1D exciton correlation function,

A(1)(τ1) = 2σI (1)A′(0)C1′ (τ1). (59)

The other 1D correlation functions cannot be observed in a1D experiment, but they can be accessed in a 2D experiment.The full 2D signal is

A(2)(τ2, τ1) = 2σσ ′I (2)A′(0)(C1′1′(τ2, τ1) − 12C2′1′ (τ2, τ1)

− 12C2′

1′1′(τ2, τ1)). (60)

Along the τ 2 = 0 axis, the 2D experiment simply duplicatesthe information in the 1D experiment

A(2)(0, τ1) = σσ ′I (2)A′(0)C1′ (τ1) = 12σ ′I2A

(1)(τ1). (61)

Along the τ 1 = 0 axis, the 2D absorbance reduces to a sum ofthe three 1D correlation functions,

A(2)(τ2, 0)=2σσ ′I (2)A′(0)(C1′(τ2)− 1

2C2′(τ2) − 12C1′

2′ (τ2))

.

(62)



FIG. 5. Two zero-time cuts through the 2D MUPPETS signal. Red (upper)curve: 2A(2)(0, τ 1), which is equivalent to the exciton decay measured in a 1Dexperiment. Blue (lower) solid curve: A(2)(τ 2, 0), which has a negative biex-citon signal superimposed on the positive exciton signal. The dashed bluecurve neglects the cross-relaxation [Eq. (53)]. The curves are normalized tothe same amplitude at long time, so the difference between these cuts mea-sures the biexciton decay [Eq. (63)].

These two cuts through the 2D signal are shown inFig. 5 as solid curves. Because they are related to 1D cor-relation functions, they contain no new information on rateheterogeneity or correlation. Nonetheless, they contain newinformation on the biexciton decay that is not available from1D measurements.

In a two-level system, these two cuts are identical.2, 8, 10

Thus, the asymmetry in τ 1 and τ 2 is diagnostic of a biexci-ton contribution to the signal. Because the two contributionshave opposite signs, the cut along τ 1 = 0 may not be mono-tonic: it can rise as the negative biexciton contribution decays.This feature is also unique to a multilevel system. The effectis weak for the parameters chosen here, but it can be moreprominent under other conditions.11, 12 It is more clearly seenin the dashed blue curve in Fig. 5, which leaves out the effectsof cross-relaxation.

This feature gives MUPPETS a unique potential to sep-arate exciton and biexciton dynamics. Subtracting the twozero-time cuts [Eqs. (61) and (62)] gives the biexciton decay

2A(2)(0, τ1) − A(2)(τ2, 0)

σσ ′I (2)A′(0)= C2′(τ2) + C1′

2′ (τ2). (63)

The small cross-relaxation term can be approximated withEq. (52) and removed.

In many systems, the exciton shift is too small to spec-trally separate exciton and biexciton dynamics. If there isa significant difference in their decay rates, 1D experimentsgive a power-dependent change in kinetics that can be iden-tified as the contribution of biexcitons. Unfortunately, a longlived photoproduct with a fast exciton decay has exactly thesame properties and can be mistaken for a biexciton.33 In a2D MUPPETS experiment, a photoproduct with a fast excitonlifetime contributes to C1′ (τ ) and is eliminated in Eq. (63).This experiment distinguishes between species that existedbefore the pulse sequence (photoproducts) and species createdduring the pulse sequence (biexcitons). This idea is illustratedin more detail by model III below (Sec. IV C 3). It will alsobe demonstrated experimentally in future papers.11, 12

This mechanism fundamentally discriminates betweenexciton and biexciton signals. If a photoproduct is presentand its biexciton decay differs from the biexciton decay ofthe primary species, the measured C2′ (τ ) will contain a mix-ture of both signals. An extrapolation to zero average poweris still needed to eliminate this possibility. The forthcom-ing papers will also explore the power dependence of theMUPPETS signal in more detail and demonstrate the neces-sary extrapolation.11, 12

The sign change between exciton and biexciton signals isdependent on having a net absorption from the exciton state(excited-state absorption minus stimulated emission) that isweaker than the absorption from the ground state. This condi-tion is satisfied in most real excitonic systems.

B. Coherent versus incoherent excitons

Any discussion of excitonic systems faces a potentialparadox. Any set of zero-order, two-level chromophores canbe grouped to form a multilevel system. To avoid a para-dox, all multiexciton effects must disappear in the absence ofa suitable interaction between the zero-order chromophores.The number of zero-order chromophores to consider is non-trivial in many systems: How many electron-hole pairs in asemiconductor? How many molecules in a dye aggregate?How many “segments” in a conjugated polymer?

First, one cannot define an excitonic system that is overlylarge. If M zero-order chromophores with an absorption cross-section σ are included, the ground-to-exciton cross-section isMσ , the exciton-to-biexciton cross-section is (M−1)σ , andso on. In the limit as M becomes large, Scheme A (Fig. 2)is reached as a limit. In this scheme, the pathways involvingmultiple excitons have zero amplitude (Fig. 3). The reason isthat absorption saturation is lost as M becomes large. Withoutnonlinear absorption, there can be no signal in a multidimen-sional experiment.

Second, one must consider the nature of the interactionbetween chromophores. In spectral correlation spectroscopy,the interaction must perturb the zero-order spectroscopy ofthe system, either splitting the transitions or transferring ab-sorption strength between exciton and biexciton transitions.This relatively strong coupling is sufficient, but not necessary,to create multiexciton effects in MUPPETS.

We focus on the more difficult case where the zero-orderspectra and cross-sections are not perturbed and an excitonwould not be seen in spectral measurements

(σD/T )0′1′(ω) = 2(σD/T )1′

2′(ω). (64)

This equation requires that both the integrated cross-sectionsand the cross-section at each frequency are not perturbed, thatis, there is no coherent coupling. Even without spectral inter-actions, there can be an interaction that perturbs the rates, forexample, one that causes exciton–exciton annihilation. Thisinteraction constitutes an incoherent coupling. This couplingexpresses itself primarily through the cross-relaxation func-tion, which we previously calculated in the limit of strongincoherent coupling, kb � 2ke, Eq. (51). In the limit of nocoupling, statistics cause the biexciton rate to be twice the



exciton rate,

kb(t) = 2ke(t), (65)

or the biexciton decay to be the square of the exciton decay,

G2′2′ (t1, t0) = (

G1′1′ (t1, t0)

)2. (66)

Putting this zero rate-coupling limit into Eq. (A8) gives

G2′1′ (t2, t1) = G1′

1′(t2, t1) − G2′2′ (t2, t1). (67)

The relevant 2D cross-relaxation function [Eq. (47)] is then

C2′1′1′(τ2, τ1) = C1′1′ (τ2, τ1) − C2′2′ (τ2, τ1). (68)

In the absence of spectral perturbations, the relativecross-sections for the three 2D pathways are those of SchemeC (Fig. 3). With Eq. (68), the cross-relaxation pathway (ii)partially cancels the exciton–exciton pathway (iii), but com-pletely cancels the exciton–biexciton pathway (i). Thus, allmultiexciton effects disappear from MUPPETS unless thereis an incoherent coupling that violates Eq. (65). Conversely,any deviation from Eq. (65) creates excitonic effects that aredetectable in MUPPETS. However, a coherent coupling thatviolates Eq. (64) is not required. Thus, a system may need tobe treated as an incoherent exciton in MUPPETS, even whenit does not need to be treated as a coherent exciton in spectralcorrelation spectroscopy.

The difference between incoherent and coherent excitonsis one of degree, not of kind. Consider the interaction energycoupling the zero-order chromophores. The inverse of this en-ergy gives an interaction time that describes the rate of energytransfer between the chromophores. To have a coherent cou-pling that is detectable in coherent spectroscopy, the interac-tion time must be on the order of or shorter than the dephasingtime, i.e., there must be coherent energy transfer. If the inter-action is weaker, it can still induce incoherent energy hoppingthat leads to exciton–exciton annihilation. So long as the anni-hilation time is on the order of or shorter than the populationdecay time, an incoherent coupling will perturb the rates andwill be detected by MUPPETS. If the population decay timeis longer than the dephasing time, a system may constitutean incoherent exciton, even when it is too weakly coupled toform a coherent exciton.

C. Measuring exciton–biexciton correlations

The full 2D-MUPPETS signal, A(2)(τ 2, τ 1) with both τ 1

and τ 2 varying, depends on correlations in the kinetics. Theexciton–exciton correlation C1′1′ (τ 2, τ 1) reports on whetherthe dispersion in C1′ (τ 1) is due to a homogeneous [Eq. (45)]or a heterogeneous [Eq. (46)] mechanism. This idea has beenthoroughly discussed in two-level systems.2, 3, 8–10 The newfeature in excitonic systems is the exciton–biexciton func-tion C2′1′(τ 2, τ 1), which reports on correlations between twodifferent transitions. To illustrate the behavior of this func-tion, we will calculate the 2D-MUPPETS signal for four lim-iting models: homogeneous or heterogeneous exciton kinet-ics combined with either correlated or uncorrelated exciton–biexciton kinetics.

The time-domain representation of the final signal foreach model is shown in Fig. 6. As discussed in Sec. IV A, the

FIG. 6. The total 2D-MUPPETS time decays A(2)(τ 2, τ 1) for models I (ho-mogeneous exciton, uncorrelated biexciton), II (heterogeneous exciton, un-correlated biexciton), III (heterogeneous exciton, correlated biexciton) andIV (homogeneous exciton, correlated biexciton). (a) The signal versus τ 1 forvarious values of τ 2 normalized at τ 1 = 0. In model I, all curves overlap.(b) The signal versus τ 2 for various values of τ 1 normalized at τ 2 = 0. Allmodels have the same 1D decays (Fig. 4).

decays in τ 1 and in τ 2 are not symmetric, a characteristic ofa multilevel system. All the models have identical 1D decays(Fig. 4), but the 2D decays in Fig. 6 are quite different. On anempirical basis, 2D MUPPETS can distinguish different lev-els of exciton heterogeneity and different levels of exciton–biexciton correlation.

A more rational discussion of the different results is pos-sible using the 2D rate spectra of the total signal and the com-ponents contributing to it (Fig. 7). In two-level systems, thediagonal of a 2D rate spectrum is always the square of the1D rate spectrum and is the same for all models.2 The spec-tra also have reflection symmetry about the diagonal. In mul-tilevel systems, these features remain in the exciton–excitoncomponents [Fig. 7(I.a–III.a) but are lost in the total spectra[Fig. 7(I.c–III.c)].

1. Model I: Homogeneous excitonand uncorrelated biexciton

In model I, all the particles are identical, i.e., there is noheterogeneity. The exciton decay of any single chromophoreis dispersed due to a complex relaxation mechanism, i.e.,the dispersion is homogeneous. In this case, the exciton–exciton correlation function in time is given by Eq. (45). Thecorresponding rate spectrum,

C1′1′ (y2, y1) = C1′(y2)C1′(y1), (69)



FIG. 7. 2D-MUPPETS rate spectra for models I (homogeneous exciton,uncorrelated biexciton), II (heterogeneous exciton, uncorrelated biexciton)and III (heterogeneous exciton, correlated biexciton). (a) The exciton–exciton component, C1′1′ (y2, y1), with y = log10(κτ 0). (b) The negativeof the exciton–biexciton component, −C2′1′ (y2, y1). (c) The total signal,A(y2, y1) ∝ C1′1′ (y2, y1) − 1

2 C2′1′ (y2, y1) − 12 C2′

1′1′ (y2, y1). Delta functionshave been broadened by a Gaussian with a width of 0.3 decades. Contoursare linear with red/orange positive, yellow zero, green/blue negative.

is shown in Fig. 7(I.a). The amplitude along the diagonal is thesquare of the 1D exciton spectrum in Fig. 4(b).2 In this model,the off-diagonal amplitude takes on its maximum value every-where. If the decays were modeled with discrete rates insteadof continuous distributions, the off-diagonal amplitude wouldappear as cross peaks linking rates lying on the diagonal.2 Theoff-diagonal amplitude shows that the corresponding diagonalrates are components of a single, complex relaxation process:the diagonal rates “co-exist” on the same chromophore.

Model I additionally assumes that the exciton and biex-citon relax by independent and unrelated mechanisms. Thus,the exciton and biexciton kinetics are uncorrelated

C2′1′(τ2, τ1) = C2′(τ2)C1′(τ1). (70)

The negative of the corresponding rate spectrum,

C2′1′ (y2, y1) = C2′(y2)C1′(y1), (71)

is shown in Fig. 7(I.b). The spectrum is no longer centered onthe diagonal, but rather on a shifted, parallel line. The spec-trum shows strong amplitude off this line, just as the exciton–exciton spectrum shows strong off-diagonal amplitude. Thus,rate homogeneity of a single transition [Eqs. (45) and (69)] isanalogous to a lack of correlation in the rates of two transi-tions [Eqs. (70) and (71)]. In either case, knowing that a rateis observed on a given chromophore in one measurement doesnot give any additional information on whether a differentrate will be observed on the same chromophore in a secondmeasurement.

The identifying characteristic of fully homoge-neous/uncorrelated kinetics is that the 2D signal is separablein the two time variables or in the two rate variables. Thisseparability extends to the cross-relaxation [Eqs. (45) and(53)] and thus, to the total signal. In the time decays ofFigs. 6(I.a) and 6(I.b), separability causes all the curves ineither plot to overlap after normalization. In the rate spectra,it is this separability that leads to a maximal spread along theanti-diagonal direction.

Figure 7(I.c) shows the rate spectrum of the total sig-nal, including the cross-relaxation. There is strong overlap ofthe exciton–exciton and exciton–biexciton components, butenough information remains to identify the important featuresof each component. A horizontal node is formed by cancella-tion between the exciton–exciton and exciton–biexciton com-ponents. The horizontal node reflects the separability of thetotal signal and, thus, is an identifying feature of a homoge-neous and uncorrelated system.

2. Model II: Heterogeneous excitonand uncorrelated biexciton

In model II, each chromophore has a simple, exponen-tial exciton decay, i.e., there is no homogeneous dispersion.The dispersion of the ensemble decay [Eq. (55)] is only dueto differences in the decay rates of different chromophores,i.e., the dispersion is due to heterogeneity. In this case, theexciton–exciton correlation function is given by Eq. (46). Thecorresponding rate-correlation spectrum,

C1′1′(y2, y1) = C1′ (y1)δ(y1 − y2), (72)

is shown in Fig. 7(II.a). The diagonal amplitude is identicalwith that of model I [Fig. 7(I.a)]. However in model II, thereis no off-diagonal amplitude. The lack of off-diagonal am-plitude indicates that different rates do not “co-exist” on asingle chromophore: each rate is associated with a differentchromophore.

As with model I, model II assumes that the exciton andbiexciton decay by independent mechanisms. In particular,the exciton heterogeneity has no effect on the biexciton decay.As a result, Eqs. (70) and (71) still hold for the biexciton–exciton correlation function, and Eq. (53) holds for the cross-relaxation. The biexciton–exciton spectrum [Fig. 7(II.b)] isunchanged from model I [Fig. 7(I.b)]. However, the totalspectrum [Fig. 7(II.c)] is quite distinct from that of model I[Fig. 7(I.c)].

The corresponding results in the time domain can be in-terpreted by regarding one time period as a rate-based fil-ter to select a subensemble whose decay is measured in theother time period. Figure 6(II.b) shows the decay in τ 2, whichmeasures the sum of exciton and biexciton decays. As τ 1

increases, the first time period progressively removes chro-mophores with a fast exciton decay. The exciton componentduring τ 2 slows as τ 1 increases. However, the biexciton com-ponent is unaffected by filtering based on the exciton decaytime. As these two components become separated in time,the signal rise due to biexciton decay becomes visibly distinctfrom the slower exciton decay.



Figure 6(II.a) shows the decay in τ 1, which measuresonly the exciton decay. When τ 2 = 0, all chromophores aremeasured. As τ 2 increases, the second time period progres-sively selects for chromophores with well-separated excitonand biexciton lifetimes, as these have less signal cancellation.With no correlation between exciton and biexciton lifetimes,these are the chromophores with a long exciton lifetime. Thus,the exciton decay in τ 1 slows as τ 2 increases.

3. Model III: Heterogeneous excitonand correlated biexciton

We now introduce exciton–biexciton correlation.Whereas lack of correlation always produces the same resultregardless of the mechanistic details, models with correlationrequire a more detailed specification of how the correlation isproduced. Model III assumes that the exciton and biexcitondecays of an individual chromophore are both exponential,that is,

G1′1′(t1, t0; θ ) = e−ke(θ)(t1−t0) (73)

and

G2′2′ (t1, t0; θ ) = e−kb(θ)(t1−t0). (74)

Dispersion in the ensemble decay is only due to heterogene-ity. In Eqs. (73) and (74), the rate is constant in time, butvaries with θ , a static or slow bath variable that varies fromchromophore to chromophore. This variable has a probabilitydistribution D(θ ), giving the 1D correlation functions

C1′(τ ) =∫

D(θ )e−ke(θ)τ dθ (75)

and

C2′(τ ) =∫

D(θ )e−kb(θ)τ dθ. (76)

As in model II, Eqs. (46) and (72) give the heterogeneousexciton–exciton time decay and rate spectrum [Fig. 7(III.a)].

In model I, the exciton and biexciton rates depended ondifferent, independent bath coordinates, ke(θ e) and kb(θb), andso their dynamics are uncorrelated. In model III, correlationoccurs because the exciton and biexciton rates depend on thesame bath variable [Eqs. (73) and (74)]. The exact nature ofthe common dependence must also be specified. For purposesof illustration, we choose

kb(θ ) = cke(θ ), (77)

which is consistent with the similarity of the excitonand biexciton decay shapes that we have already assumed[Eq. (57)]. The biexciton–exciton correlation function[Eq. (47)],

C2′1′ (τ2, τ1) =∫

D(θ )G2′2′(t2, t1; θ )G1′

1′(t1, t0; θ ) dθ, (78)

reduces to

C2′1′(τ2, τ1) = C1′(cτ2 + τ1) = C2′ (τ2 + τ1/c). (79)

When c = 1, this equation reduces to the exciton–excitonresult for pure heterogeneity [Eq. (46)]. Thus, pure hetero-geneity on a single transition is analogous to perfect correla-

tion between two transitions. In a purely heterogeneous sam-ple, one measurement of the exciton rate on a chromophoregives perfect knowledge of the biexciton rate that will befound in a subsequent measurement.

The negative of the corresponding exciton–biexciton ratespectrum,

C2′1′ (y2, y1) = C1′(y1)δ(y1 − y2 + ln c), (80)

is shown in Fig. 7(III.b). The spectrum traces out a curvein the y2−y1 plane. With the simple correlation defined byEq. (77), the curve is a straight line. Others forms would gen-erate more complex curves. In general, an experimental resultin the form of a one-dimensional curve is diagnostic for corre-lated heterogeneity, and the form of the curve allows the formof the correlation to be inferred.

The total rate spectrum and time decay are shown inFig. 7(III.c) and Fig. 6(III.a and III.b), respectively. These in-clude the cross-relaxation,

C2′1′1′ (τ2, τ1) = C1′ (cτ2 + τ1) − C1′((c + 1)τ2 + τ1), (81)

which is calculated from Eqs. (49) and (51), and its ratespectrum,

C2′1′1′ (y2, y1) = C1′(y1)[δ(y1 − y2 + ln c)

− δ(y1 − y2 + ln (c + 1))]. (82)

In this figure, the node of the rate spectrum lies parallel tothe diagonal, reflecting the simple linear form of Eq. (77).More generally, the node will reflect the shape of the exciton–biexciton correlation function and, thus, the form of the cor-relation.

The interpretation of the time decays is similar to that formodel II. In Fig. 6(III.b), as τ 1 increases, chromophores withfast relaxing excitons are eliminated from the measurement.In this model, the remaining chromophores have both a slowerexciton and a slower biexciton decay. Both the rise and fall ofthe signal are delayed as τ 1 increases. Figure 6(III.a) showsthe converse effect. As τ 2 increases, only chromophores withslow decays (either exciton or biexciton) reach the detectionphase of the experiment. The exciton decay of the selectedchromophores is measured during τ 1 and slows as the selec-tion criterion becomes stricter.

4. Model IV: Homogeneous excitonand correlated biexciton

Model I considered the case of purely homogeneous dis-persion in the exciton and biexciton decays. More precisely,each chromophore had a time dependent rate ke(t) and kb(t) forthe exciton and biexciton, respectively. Underlying this time-dependence is a bath variable ϕ(t) that is relaxing to a newvalue in the excited state. In model I, the exciton and biexci-ton rates depend on different, independent bath coordinates,ke(ϕe(t)) and kb(ϕb(t)), and so their dynamics were uncorre-lated. Model IV makes the same basic assumptions,

C1′ (τ1) = G1′1′(t1, t0; ϕ) = exp

(−

∫ t1

t0

ke(ϕ(t))dt

)(83)



and

C2′ (τ1) = G2′2′ (t1, t0; ϕ) = exp

(−

∫ t1

t0

kb(ϕ(t))dt

), (84)

but assumes that the exciton and biexciton decays depend onthe same bath property, and so are perfectly correlated.

In the absence of heterogeneity, the exciton–exciton cor-relation function is the same as in model I [Eqs. (45) and (69)].The biexciton–exciton correlation function is calculated with-out ensemble averaging, i.e., from

C2′1′ (τ2, τ1) = G2′2′ (t2, t1)G1′

1′(t1, t0), (85)

but more information on the dynamics of ϕ(t) is needed. Wemake the simple assumption that the dynamics of ϕ(t) are thesame in the exciton and biexciton state. In this case,

C2′1′ (τ2, τ1) = G2′2′ (t2, t0)G1′

1′ (t1, t0)

G2′2′ (t1, t0)

= C2′(τ2 + τ1)C1′(τ1)

C2′(τ1). (86)

This result can be interpreted by writing it as

C2′1′(τ2, τ1) = (1 + Z(τ2, τ1)) C2′ (τ2)C1′(τ1) (87)

with

Z(τ2, τ1) = C2′(τ2 + τ1)

C2′(τ2)C2′(τ1)− 1. (88)

The function Z(τ 2, τ 1) measures the rate dispersion ofC2′ (τ ). When C2′ (τ ) is an exponential, Z(τ 2, τ 1) = 0 every-where. When C2′(τ ) is not exponential, Z(τ 2, τ 1) is still zeroalong the τ 1 = 0 and τ 2 = 0 edges of its domain, but it isnonzero in the middle: positive if the rate slows with time, andnegative if the rate increases with time. Thus, Eq. (87) has themaximum deviation from the uncorrelated result [Eq. (45)]allowed by the dispersion of C2′ (τ ). For our model functions,this deviation is a positive one for large values of τ 1 and τ 2.Under certain conditions, this deviation can give a signal thatrises with delay in some regions, for example, in Fig. 6(IV.b).Rate spectra for this model are difficult to calculate and arenot easy to interpret and so are not presented.

V. THERMAL SIGNALS IN MULTILEVEL SYSTEMS

A. General formalism

Heterodyned experiments are not only sensitive to res-onant absorption from the solute; they are also sensitive toindex-of-refraction changes in the solvent due to the heat re-leased by nonradiative decay. In 1D, these effects are calledthermal gratings.13–15 (The total thermal response can be sep-arated into a pure thermal and an acoustic component, but thatdistinction will not be needed here.) In Ref. 7, we showedhow to incorporate thermal effects into pathway calculationsof multidimensional experiments. Here that treatment is ex-tended to multilevel systems.

The system states must be expanded to include not onlythe electronic state of the solute P, but also the energy densityof the solvent ε, that is, the state must have the form |P ε]. Theenergy density is measured at the same (suppressed) k-vector

FIG. 8. Pathways for the calculation of thermal signals in one-dimensional(1D) and two-dimensional (2D) experiments [see Fig. 3]. The final two statesof the pathways are expanded to |P nε] to show both P, the electronic state,and nε , the number of quanta of thermal energy deposited in the solvent.

as the electronic state. The response to the solvent energy islinear, so |P ε1] + |P ε2] = |P ε1 + ε2]. It will be convenient toshift from ε, the heat per volume of solvent, to nε, the numberof photons of energy converted to heat per solute molecule,

nε = ε

¯ωρ. (89)

An important result of Ref. 7 is that in a multidimensional ex-periment, only the thermal signal formed by the last excitationis detectable. Thus, the expanded states are only needed at theend of the pathways (see Fig. 8).

The generalized absorption due to thermal effectsA(N)

ε (τN, . . . , τ1) adds to the resonant absorption A(N)(τN, . . . ,τ 1) [Eq. (9)] and can be expressed in an analogous form,

A(N)ε (τN, . . . , τ1) = (−1)N ρLI (N) [I |σDε

×∫ tN

tN−1

Cε(tN − t ′)d

dt ′Gε(t ′, tN−1) dt ′

× . . . σT G(t1, t0)σT |eq]. (90)

The thermal detection cross-section operator σDε can be ex-pressed in terms of nε, the operator that measures the valueof nε,

σDε = iσ ′′ε nε. (91)

Because the thermal response is a change in the index-of-refraction, this operator is imaginary. Its magnitude is

σ ′′ε = ω

c

(1 + 1

n2s

) (−dns

dρs

) (dρs

dεs

)¯ω, (92)

where ns is the solvent index-of-refraction and ρs is the sol-vent density. This quantity has the units of a cross-section andis normally real and positive. The time-evolution operator forthe electronic state G(t ′, t) is expanded to Gε(t ′, t), the time-evolution operator of the combined electronic-thermal state,for the last time period.

The detection is not of the energy itself, but of the re-sulting change in index-of-refraction. In Eq. (90), the en-ergy deposition is convolved with Cε(τ ), the time-evolutionof thermal energy into an index-of-refraction change. Sophis-ticated expressions for Cε(τ ) valid over a wide time rangeare available.14, 34–37 For purposes of illustration over short



times,

Cε(τ ) = 1 − cos(2τ/�) (93)

is an adequate expression.7 This thermal correlation functionis zero when τ = 0 and reaches a maximum of two at half theacoustic period � due to interference between the slowly de-caying pure thermal response and the more rapidly oscillatingacoustic response.

The convolution in Eq. (90) can be removed, if the de-cay of the electronic state is much faster than the acoustic pe-riod. If the decay is not complete within the acoustic period,but only times ��/2 are treated, this approximation can bepushed farther. The fraction that decays before �/4 (halfwayto the maximum) is treated as decaying instantaneously, andthe fraction that decays after �/4 is treated as never decaying.This approximation is rough when the solute relaxation has asingle timescale, but becomes more reasonable when the de-cay is highly dispersed in time. In this approximation,

A(N)ε (τN, . . . , τ1) = (−1)N ρLI (N) [I |σDεCε(τN )

× Gε(�/4 + tN−1, tN−1) . . .

×σT G(t1, t0)σT |eq]. (94)

The primed basis set for electronic states can be intro-duced for the thermal pathways, as they were for resonantpathways in Sec. II B. The thermal absorption is then writ-ten [compare to Eq. (24)]

A(N)ε (τN, . . . , τ1)

A′(0)= I (N) (σε)0′p,...,l′,j ′

m′0,...,k′

× (Cε)m′0,...,k′,1′

0′p,...,l′,j ′ (τN, . . . , τ1). (95)

The final two indices are expanded to include the thermal vari-ables. The total thermal cross-section is given by [compare toEq. (25)]

(σε)n′p,...,l′,j ′

m′0,...,k′ = (−1)N (σDε)n′p (σT )l

′m′0 . . . (σT )j

′k′ . (96)

The full operator σDε has been reduced by one dimension andconverted to a vector as in Eq. (22),

[σDε| = (σT )0′1′

Re (σD)0′0′

[0′p|σDε, (97)

with the result that

(σDε)n′p = i

√2δn′0′σ ′′

ε p. (98)

Because σDε is diagonal in the electronic state, only the n′

= 0 elements are nonzero. The multidimensional correla-tion function in Eq. (95), which corresponds to the one inEq. (14), is

(Cε)m′0,...,i ′

0′p,...,j ′ (τN, . . . , τ1) = Cε(τN )

× ⟨(Gε)m

′00′p (�/4 + tN−1, tN−1) . . . Gi ′

j ′ (t1, t0)⟩. (99)

The time evolution in the last time period is now governed bythe thermal response, rather than by solute dynamics.

B. Results for excitonic systems

In an excitonic system, the number of pathways isseverely limited. As with the electronic signal, the primedbasis set yields the minimum number of pathways. Figure 8shows the allowed pathways for N = 1 and N = 2. Only twoelements of Gε(t′, t) are needed. In calculating them, we al-low nonradiative decay that leads to long lived, high energystates (“trap” states) without the immediate release of heat.The fractional yield of heat for the biexciton-to-exciton andexciton-to-ground transitions are Q2 and Q1, respectively. Therequired matrix elements are then

(Gε)1′00′1(t ′, t) = Q1√

2

(1 − G1′

1′ (t ′, t)),

(100)

(Gε)2′00′1(t ′, t) = Q2√

2

(1 − G2′

2′ (t ′, t)) + Q1√

2G2′

1′ (t ′, t).

In the primed basis set when the cross-relaxation is small,each thermal pathway is dominated by the relaxation of a sin-gle electronic transition.

Combining Eqs. (95)–(100) with the pathways in Fig. 8yields expressions for the thermal signals,

A(1)ε (τ1) = A′(0)I (1)(−iσ ′′

ε )Cε(τ1)Q1 (1 − C1′ (�/4)) (101)

and

A(2)ε (τ2, τ1) = A′(0)I (2)(−iσ ′′

ε )(2σ ′)Cε(τ2)

×{ (Q1 − 1

2Q2)C1′(τ1) − Q1C1′1′ (�/4, τ1)

+ 12

[Q2C2′1′(�/4, τ1) − Q1C

2′1′1′ (�/4, τ1)

]}.

(102)

The results for the different models in Fig. 2 differ in onlyminor ways; model C has been used for specificity. The 1Dresult is consistent with previous work.13–15 The 2D result isnew. It allows the thermal effects to be calculated from thecorrelation functions already discussed in Sec. IV. The ther-mal cross-section in the 2D expression can be obtained from1D experiments. The only new information in the 2D ther-mal signal is the quantum yield of heat for the biexciton de-cay. Thus, 2D experiments have the potential to measure thisquantity.

VI. CONCLUSIONS

This paper has laid the theoretical foundation for MUP-PETS in multilevel systems, especially excitonic systems. Thecalculations were simplified by introducing a nonorthogonalbasis set. By using population conservation, the number ofstates to be considered was reduced by one. In an excitonicsystem, the number of pathways and correlation functionsare reduced further. An unavoidable complication of multi-level systems is cross-relaxation between basis states. How-ever, suitable approximations were found in the limits of ei-ther strong or weak exciton–exciton interaction. Methods forcalculating thermal effects in multilevel systems were alsopresented.

Using these methods, the new information available fromMUPPETS was demonstrated. MUPPETS was shown to



be very sensitive to chromophore interactions. First, it wasshown that much weaker interactions are needed to observekinetic effects, that is, to form an incoherent exciton, than areneeded to observe spectral effects, that is, to form a coherentexciton. In an incoherent exciton, chromophores interact byincoherent energy hopping followed by exciton–exciton an-nihilation. Second, it was shown that MUPPETS is a sensi-tive method for detecting incoherent exciton formation. Anyasymmetry in the decays along the two time axes is a sign ofan incoherent exciton. The difference between these decaysis a direct route to the biexciton decay rate and, thus, to thestrength of exciton–exciton interactions. Exciton–exciton an-nihilation can also be measured by power-dependent 1D ex-periments, but these measurements can be confounded by thebuild-up of long-lived photoproducts with short exciton life-times. MUPPETS is immune to this problem.

Away from the time axes, MUPPETS offers additionalinformation for systems with rate dispersion. Both excitonrate heterogeneity and correlations between exciton and biex-citon dynamics are available. Example calculations suggestthat there is sufficient information to allow a unique separa-tion of these two effects in most cases. Rate heterogeneity is aconcept that has been explored in previous MUPPETS studiesof two-level system; the concept of correlated rates betweentwo transitions is a new one. When the rates of two transitionsare correlated, the MUPPETS results are similar to those forheterogeneous rates on a single transition. Correlation indi-cates that the relaxation mechanisms of the two transitionsare linked. Correlation is possible whether the individual re-laxations are heterogeneous or homogeneous. In the hetero-geneous case, individual particles relax either faster or slowerthan average for both transitions. In the homogeneous case,the relaxations of both transitions depend on the relaxation ofa common bath mode.

The practicality of these ideas will be demonstrated infuture papers.11, 12 The results in this paper provide a basisfor both a qualitative and quantitative interpretation of thoseresults.

ACKNOWLEDGMENTS

This material is based upon work supported by theNational Science Foundation (NSF) under CHE-1111530.

APPENDIX: OFF-DIAGONAL TIME EVOLUTION

The calculation of the off-diagonal elements of theGreen’s function starts by dividing the time evolution betweentwo times, t1 and t2, by M intermediate times t′a

G(t2, t1) = G(t2, t′M ) . . . G(t ′a+1, t

′a) . . . G(t ′1, t1). (A1)

Taking matrix elements gives

G2′1′ (t2, t1) = Gn′

1′ (t2, t′M ) . . . Gk′

l′ (t ′a+1, t′a) . . . G2′

i ′ (t′1, t0),

(A2)where the indices i, . . . , n run over all nonzero states. Becauserelaxation is only downward, all but one of these matrix ele-

ments must be diagonal. The only remaining terms are

G2′1′ (t2, t1) =

N−1∑a=1

G1′1′(t2, t

′a+1)G2′

1′(t ′a+1, t′a)G2′

2′(t ′a, t1), (A3)

where sequences of diagonal elements have been recombined.The limit M → ∞ and dt′ = t′a+1− t′a → 0 can now be ap-plied. Equation (7) provides the infinitesimal Green’s operator

G(t + dt ′, t) = 1 − R(t)dt ′, (A4)

resulting in

G2′1′ (t2, t1) = −

∫ t2

t1

G1′1′ (t2, t

′)R2′1′ (t ′)G2′

2′(t ′, t1)dt ′. (A5)

Using Eq. (32) for the rate matrix element gives Eq. (50) ofthe main text.

We now use the specific structure of an excitonic rate ma-trix [Eq. (32)] to replace the off-diagonal rate with a diagonalelement

G2′1′ (t2, t1) =

∫ t2

t1

G1′1′ (t2, t

′)R1′1′ (t ′)G2′

2′(t ′, t1)dt ′. (A6)

Because relaxation is only downward, Eq. (7) also applies todiagonal matrix elements and yields

G2′1′ (t2, t1) =

∫ t2

t1

(d

dt ′G1′

1′(t2, t′))

G2′2′(t ′, t1)dt ′. (A7)

Integration by parts gives

G2′1′(t2, t1) = G2′

2′ (t2, t1) − G1′1′(t2, t1)

−∫ t2

t1

G1′1′ (t2, t

′)(

d

dt ′G2′

2′ (t ′, t1)

)dt ′. (A8)

This form can be used directly to derive Eq. (67) in the limitof zero incoherent coupling [Eq. (66)].

To look at the opposite limit of strong coupling, we definea change in occupation of |1′],

δG1′1′ (t, t1) = 1 − G1′

1′ (t, t1), (A9)

which is assumed to be small over the biexciton lifetime. Theterm in Eq. (A8) can be written

G1′1′(t2, t) = G1′

1′ (t2, t1)

1 − δG1′1′ (t, t1)

. (A10)

Putting a power series expansion of Eq. (A10) into Eq. (A8)and integrating the first term leads to

G2′1′ (t2, t1) = G2′

2′ (t2, t1) − G1′1′(t2, t1)

[G2′

2′ (t2, t1)

+∫ t2

t1

δG1′1′ (t ′, t1)k2(t ′)dt ′

+1

2

∫ t2

t1

(δG1′

1′ (t ′, t1))2

k2(t ′)dt ′ + . . .

]. (A11)

Keeping only the leading term gives Eq. (51) of the main text.The same results hold if the states 2′ and 1′ are replaced byany two neighboring states.



We note that a simple, empirical formula interpolates be-tween the limits of strong [Eq. (51)] and zero [Eq. (67)] inco-herent coupling

G2′1′ (t2, t1) = G2′

2′(t2, t1)

G1′1′(t2, t1)

(1 − G1′

1′ (t2, t1)). (A12)

The accuracy of this approximation has not been tested.One can consider couplings outside this range. In this

case, the biexciton decay rate is less than twice the excitondecay rate. The presence of a second excitation slows the de-cay of the first. Although this situation is not forbidden, it isuncommon.

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